In this chapter, you will learn the basic of stabilizer states, and Clifford Maps

2. State-Map Duality

Every stabilizer state \(\rho\) is dual to a Clifford unitary \(U\), such that the state can be generated from the zero state \(|00\cdots0\rangle\) as $\(\rho = U|00\cdots0\rangle\langle 00\cdots0|U^\dagger\)\(. Both \)\rho\( and \)U$ describes a stabilizer code:

• \(\rho\) is a projection operator that specifies the code subspace of the stabilizer code.

• \(U\) is the encoding Clifford unitary that encodes the logical + syndrome qubits to the physical qubits in the stabilizer code.

The package stabilizer (based on paulialg) provides related functions to represent stabilizer states and Clifford maps. There are two classes defined in this package.

• stabilizer.CliffordMap. Since the Clifford unitary \(U\) maps Pauli operators to Pauli operators, it is sufficient to specify a Clifford unitary by how each single-qubit Pauli operator transforms under the unitary. Such transformation rules are stored in a table called the Clifford map.

• stabilizer.StabilizerState. The stabilizer state is specified by a set of stabilizers and the corresponding destabilizers. Using the binary representation of Pauli operators, they can be stored in a table, called the stabilizer tableau.

Since both classes need to store a table of Pauli operators, they are both realized as subclasses of paulialg.PauliList.